On a convergence of positive continuous additive functionals in terms of the smooth measures

Abstract A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals (PCAFs) is derived in terms of their smooth measures. To this end, we first introduce a metric on the space of measures of finite energy integrals and show some structures of the metric. Then, we show the compactness and give some examples of PCAFs that the convergence holds in terms of the associated smooth measures.